Cyclic sum of certain parametrized multiple series

In the present paper, we prove the cyclic sum formulas for certain parametrized multiple series.     

Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind

Recently, the authors introduced some generalizations of the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials (see [Q.-M. Luo, H.M. Srivastava, J. Math. Anal. Appl. 308 (2005) 290-302] and [Q.-M. Luo, Taiwanese J. Math. 10 (2006) 917-925]). The main object of this paper is to investigate an analogous generalization of the Genocchi polynomials of higher order, that is, the so-called Apostol-Genocchi polynomials of higher order. For these generalized Apostol-Genocchi polynomials, we establish several elementary properties, provide some explicit relationships with the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials, and derive various explicit series representations in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) zeta function. We also deduce their special cases and applications which are shown here to lead to the corresponding results for the Genocchi and Euler polynomials of higher order. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called λ-Stirling numbers of the second kind, we derive some basic properties and formulas and consider some interesting applications to the family of the Apostol type polynomials. Furthermore, we also correct an error in a previous paper [Q.-M. Luo, H.M. Srivastava, Comput. Math. Appl. 51 (2006) 631-642] and pose two open problems on the subject of our investigation.     

Riemann zeta function and Lyapunov-type inequalities for certain higher order differential equations

This note generalizes the well known Lyapunov-type inequalities for second-order linear differential equations to certain 2M-th order linear differential equations with five types of boundary conditions. The usage of the best constant of some Sobolev-type inequalities clarify the process for obtaining such inequality and sharpen the result of Çakmak [2].     

The Ihara zeta function of the complement of a semiregular bipartite graph

We study the Ihara zeta function of the complement of a semiregular bipartite graph. A factorization formula for the Ihara zeta function is derived via which the number of spanning trees is computed. For a class of complements of semiregular bipartite graphs, it is shown that they have the same Ihara zeta function if and only if they are cospectral.     

An identity of Andrews, multiple integrals, and very-well-poised hypergeometric series

We give a new proof of a theorem of Zudilin that equates a very-well-poised hypergeometric series and a particular multiple integral. This integral generalizes integrals of Vasilenko and Vasilyev which were proposed as tools in the study of the arithmetic behaviour of values of the Riemann zeta function at integers. Our proof is based on limiting cases of a basic hypergeometric identity of Andrews. Dedicated to Richard Askey on the occasion of his 70th birthday. Research partially supported by the programme “Improving the Human Research Potential” of the European Commission, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”. 2000 Mathematics Subject Classification Primary—33C20; Secondary—11J72     

On the zeros ofζ (1)(s) — a (on the zeros of a class of a generalized Dirichlet series — XVII)

Some very precise results (see Theorems 4 and 5) are proved about thea-values of thelth derivative of a class of generalized Dirichlet series, forl≥l _o =l _o(a) (l _o being a large constant). In particular for the precise results on the zeros ofζ ⁽¹⁾ (s) —a (a any complex constant andl≥l _o) see Theorems 1 and 2 of the introduction.     

Convergence acceleration of series through a variational approach

By means of a variational approach we find new series representations both for well-known mathematical constants, such as π and the Catalan constant, and for mathematical functions, such as the Riemann zeta function. The series that we have found are all exponentially convergent and provide quite useful analytical approximations. With limited effort our method can be applied to obtain similar exponentially convergent series for a large class of mathematical functions.     

Discrete Universality of L-Functions for New Forms

A Voronin-type discrete universality theorem for the L-functions of new forms is proved.     

Irrationality of the sums of zeta values

In this paper, we establish a lower bound for the dimension of the vector spaces spanned over ? by 1 and the sums of the values of the Riemann zeta function at even and odd points. As a consequence, we obtain numerical results on the irrationality and linear independence of the sums of zeta values at even and odd points from a given interval of the positive integers.     

An Approximate Functional Equation for the Lerch Zeta Function

In this paper, we establish an approximate functional equation for the Lerch zeta function, which is a generalization of the Riemann zeta function and the Hurwitz zeta function.